What makes expectation values real?

[TEFLing]

If you have some wave function of some particle, say...

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And you calculate the expectation value of momentum, say...

<¥|p|¥>

What ensures that that spatial integral is real valued?

Separately, all the components of the integral are complex valued

 

[vanhees71]

The operators, representing observables, are usually assumed to be self-adjoint, i.e., $\hat{p}=\hat{p}^{\dagger}$. Now you have
$$\langle \psi|\hat{p} \psi \rangle^*=\langle \psi|\hat{p}^{\dagger} \psi \rangle = \langle \psi|\hat{p} \psi \rangle,$$
which implies that the expectation value is real.